Central Limit Theorem

5.4 Central Limit Theorem

The central limit theorem (CLT) is one of the most powerful theorems within statistics. The central limit theorem: Let X₁,X₂,…,X_n be a random sample from i.i.d. random variables from any distribution with finite mean, μ, and finite variance, σ². Then the limiting distribution of
$[ ] lim X¯n--√--μ- ~ N (0, 1), n→∞ σ∕ n$ (5.3)

where _n is the average of the n sampled observations. That is for a sufficiently large sample size, n, the sample mean from i.i.d. random variables with a finite mean and finite variance has an approximately Normal distribution N(μ,σ²∕n) and X¯n-√-μ
σ∕ n is approximately N(0, 1). In real life σ is almost almost always unknown, but we can calculate a sample variance, s². If is from a random sample, X₁,X₂,…,X_n, from a normal distribution with mean μ, and finite variance, σ² then

√ -- tn- 1 = ¯x---μ--= x¯-√-μ-= --n-(x¯---μ) s.e.(¯x) s ∕ n s

has a t-distribution with d.f. = n - 1, where d.f. stands for degrees of freedom. For a sufficiently large sample size, n, from any distribution with finite mean and variance

¯x--√-μ- s∕ n ~ tn-1.

Typically a minimum of n > 30 is desired before assuming a t- distribution when the data are known to come from a non-normal distribution. The t-distribution converges to the normal distribution as n →∞, i.e.

limn → ∞ [tn- 1] ~ N (0,1).

Figure 5.1: Histogram of from population Table ?? for SRSWOR of sample size n = 1.

Figure 5.2: Histogram of from population Table ?? for SRSWOR of sample size n = 2.

Figure 5.3: Illustrating the Power of the Central Limit Theorem.

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